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ESSENTIAL QUESTION

How can we use exponents to understand fucntions?

Common Core Standards 2017-2018 Print Unit 6
High School - Math - Algebra I - Unit 6
Exponential Functions
* Multiply and divide expressions with exponents * Understanding zero and negative exponents * Comparing and contrasting growth and decay problems
OVERVIEW Show All | Hide All | Top

Students will explore exponential functions by multiplying and dividing expressions with exponents.  They will also learn how to graph these functions through exponential growth and exponential decay while applying to real-life functions.

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COMMON CORE STANDARDS Show All | Hide All | Top

-Properties of integer exponents 

(N.RN.2, A.SSE.2, A.SSE.3c)

-Rational exponents, applying properties of integer exponents (N-RN.1, N-RN.2, N-RN.3)

-Define, evaluate and compare exponential functions; identify domain, range and intercepts (F.IF.1, F.IF.5, 
  F.IF.7e)

• Classify exponential growth and decay. (F.IF.8b, F.LE.1c)

• Create and graph a function modeling two quantities. (F.IF.4, F.LE.2, F.BF.1)

   Geometric sequences (F.IF.3, F.BF.2)


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STUDENT LEARNING TARGETS Show All | Hide All | Top

N.RN.1

I can define radical notation as a way to represent rational exponents.

I can explain the properties of operations of rational exponents as an extension of the properties of integer exponents.

I can explain how radical notation, rational exponents and properties of integer exponents relate to one another.

N.RN.2

I can rewrite a radical expression using rational exponents.

I can rewrite an expression with a rational exponent using a radical expression.

N.RN.3

I can find the sums and products of rational and irrational numbers.

I can recognize that the sum of a rational number and an irrational number is irrational.

I can recognize that the product of a nonzero rational number and an irrational number is irrational.

I can explain why rational numbers are closed under addition or multiplication.

A.SSE.2

I can identify various structures of expressions.

I can use the structure of an expression to identify ways to rewrite it.

I can classify expressions by structure and develop strategies to assist in classification.

A.SSE.3c

I can define an exponential function .

I can rewrite exponential functions using the properties of exponents.

I can rewrite exponential expressions to identify properties of quantities represented by the original expression.

I can explain the properties of the quantities represented by the transformed exponential expression.

F.IF.1

I can identify the domain and range of a function.

I can determine if a relation is a function.

I can determine the value of the function with proper notation (i.e. f(x)=y, the y value is the value of a function at a particular value of x.)

I can evaluate functions for given values of x.

F.IF.3  

I can recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.  For example, the Fibonacci sequence is defined recursively by f(0)=f(1)=1, f(n+1)=f(n)+f(n-1) for n> 1.

F.IF.4

I can define and recognize the key features in tables and graphs of linear and exponential functions.

I can identify whether the function is linear or exponential, given its table or graph.

I can interpret key features of graphs and tables of functions.

I can create a graph that models the description and indicates all of the key features of the function.

F.IF.5

I can identify and describe the domain of a function when given a graph or description of the function.

I can identify an appropriate domain based on the unit, quantity, and type of function it describes.

I can relate the domain of the function to its graph.

I can explain why a domain is appropriate for a given situation.

F.IF.7e

I can graph exponential functions, by hand in simple cases.

I can graph more complicated exponential functions using technology, showing intercepts and end behavior.

F.IF.8b

I can distinguish between exponential functions that model exponential growth and exponential decay.

I can use the properties of exponents to evaluate expressions for exponential functions in a real-world

F.LE.1c

I can identify situations that display equal ratios of change over equal intervals and can be modeled by exponential functions.

F.LE.2

I can recognize that geometric sequences can be expressed as exponential functions.

I can construct exponential functions, including geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

F.BF.1

I can define “explicit function” and “recursive process”.

I can identify the relationship between quantities in a real-world problem.

I can apply algebraic transformations of parent functions.

I can write a function that describes a real-world problem.

F.BF.2

I can identify geometric patterns in given sequences.

I can generate geometric sequences from recursive and explicit formulas.

I can use given and constructed geometric sequences to model real-life situations.

I can determine the recursive rule given geometric sequences. 

I can justify the translation between the recursive form and explicit formula for geometric sequences.

I can write an explicit formula for a geometric sequence.

 

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CRITICAL VOCABULARY Show All | Hide All | Top

exponential fucntion, exponential growth and decay, percent of increase and decrease, product of powers property, power of a power property, and power of a product property

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LEARNING EXPERIENCES Show All | Hide All | Top
Educator Uploaded Plans (These are educators specific templates with included information and specifics)
E-TOOLS Show All | Hide All | Top

Coming Soon.

RESOURCES Show All | Hide All | Top
common core, textbook, khanacademy.com, Ti-84 graphing calculator, geometry sketchpad, zoom math,
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LITERACY STRATEGIES Show All | Hide All | Top

Thinking Strategies for Readers

Researchers who have studied the thinking processes of proficient readers conclude that if teachers taught the following strategies instead of much of the traditional skills-based reading curriculum, students who use the strategies would be better equipped to deal with a variety of texts independently (Keene and Zimmerman, 1997). These strategies are use­ful for composing meaning at both a text and word level.

Monitoring for Meaning

at a text level, readers . . .

 pause to reflect on their growing understandings

 recognize when they understand the text, and when they don’t

 identify when and why the meaning of the text is unclear

 identify the ways in which a text becomes gradually more understandable by reading past an unclear portion and by rereading text

 decide if clarifying a particular confusion is critical to overall understanding

 explore a variety of means to remedy confusion

 consider, and sometimes adjust, their purpose for reading

 check, evaluate and make revisions to their evolving interpretation(s) of text

at a word level, readers . . .

 identify confusing words

 employ a range of options for reestablishing meaningful reading (e.g., rereading, reading on, using words around the unknown word, using letters and sounds, using a meaningful substitution)

 

Activating, Utilizing and Building Background Knowledge (Schema)

at a text level, readers . . .

 activate relevant, prior knowledge before, during and after reading

 build knowledge by deliberately assimilating new learning with their related prior knowledge

 clarify new learning by deleting inaccurate schema

 relate texts to their world knowledge, to other texts and to their personal experiences

 activate their knowledge of authors, genre, and text structure to enhance understanding

 recognize when prior knowledge is inadequate and take steps to build knowledge necessary to understand

at a word level, readers . . .

 apply what they know about sounds-letter relationships and word parts to make sense of unknown words

 

Asking Questions

at a text-level, readers . . .

 generate questions before, during and after reading about the text’s content, structure and language

 ask questions for different purposes, including clarifying their own developing understandings, making predictions, and wondering about the choices the author made when composing

 realize that one question may lead to others

 pursue answers to questions

 consider rhetorical questions inspired by the text

 distinguish between questions that lead to essential/deeper understandings and “just curious” types of questions

 allow self-generated questions to propel them through text

 contemplate questions posed by others as inspiration for new questions

at a word level, readers . . .

 pose self-monitoring questions to help them understand unknown/unfamiliar words (e.g., “What would make good sense?”, “What would sound like language?”, “What would sound right and match the letters?”, “Is this a word I want to use as a writer? If so, how am I going to remember it?”)

 

Drawing Inferences

at a text level, readers . . .

 draw conclusions about their reading by connecting the text with their schema

 make, confirm, and/or revise reasonable predictions

 know when and how to infer answers to unanswered questions

 form unique interpretations to deepen and personalize reading experiences

 extend their comprehension beyond literal understandings of the printed page

 make judgments and create generalizations about what they read

 create a sense of expectation as they read

at the word level, readers . . .

 use context clues and their knowledge of language to predict the pronunciation and meaning of unknown/unfamiliar words

 

Determining Importance

at a text level, readers . . .

 identify key ideas, themes and elements as they read

 distinguish between important and unimportant information using their own purpose(s), as well as the text structures and word cues the author provides

 use text structures and text features to help decide what is essential and what is extraneous

 use their knowledge of important and relevant parts of text to prioritize what they commit to long-term memory and what they retell and/or summarize for others

 consider the author’s bias/point of view

 use the filter of essential/other to clarify usefulness when applying other cognitive strategies to their reading

at a word level, readers . . .

 determine which words are essential to the meaning of the text

 know when choosing to skip words/phrases of text will or will not impact their overall un­derstanding

 make decisions about when unknown/unclear words need clarification immediately and accurately, and when substitutions can be used to maintain meaning and fluency

Creating Sensory Images

at a text level, readers . . .

 immerse themselves in rich detail as they read

 create images connected to the senses of sight, hearing, taste, touch and smell to enhance and personalize understandings

 attend to “heart” images – feelings evoked while reading

 revise their images to incorporate new information and new ideas revealed in the text

 adapt their images in response to the images shared by other readers

at a word level, readers . . .

 use visual, auditory and kinesthetic modes when learning how words work

 use what they know about a word’s appearance (e.g., length, spacing above and below the line) to understand unknown words

 ask themselves “Does that look right?” and “Does that sound right?” whencross-checking unknown words

 

Synthesizing Information

at a text level, readers . . .

 continually monitor overall meaning, important concepts and themes while reading

 recognize ways in which text elements fit together to create larger meaning

 create new and personal meaning

 develop holistic and/or thematic statements which encapsulate the overall meaning of the text

 capitalize on opportunities to share, recommend and criticize books

 attend to the evolution of their thoughts across time while reading a text, and while reading many texts

at a word level, readers . . .

 select specific vocabulary from the text(s) to include in their synthesis because they know that specific language is highly meaning-laden

 know when certain vocabulary is critical to the text’s overall meaning, and therefore, must be understood if comprehension is to be achieved

 

Problem Solving

at a text level, readers . . .

 know that once meaning has broken down, that any of the other cognitive behaviors can be employed to repair understanding

 use information from the three deep surface structure systems to repair text meaning

at a word level, readers . . .

 use information from the three surface structure systems to solve word issues

 select from a wide range of word strategies (e.g., skip and read on, reread, use context clues, use the letters and sounds, speak to a peer reader) to help make sense of unknown words

 develop reading fluency

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THOUGHTFUL EDUCATION TOOLS Show All | Hide All | Top

Thoughtful Strategies by Learning Style 

Mastery

Interpersonal

Understanding

Self – Expressive

Utility (Can be used in multiple styles)

Fact or Fiction

Spider/Fist List

Word Association

Word Wall

Reading for Meaning

Interactive Lecture

Group & Labeling

Categories

Memory Box

Write to Learn

Building Writing

Reciprocal Learning

Think/Pair/Share

Give one, Get one

Collaborative Summarizing

Jeopardy

Anticipation Guides

KWL

Concept Attainment

Compare/Contrast

1,2,3,4

Yes, No, Why

Etch-a-Sketch

Mystery

Graduated Difficulty

Comprehension Menu

Task Rotation

Voc Notebook

Carousel Brainstorming

Boggle

Reader’s Theatre

Vocabulary Code

TGT

Jigsaw

4-2-1 Free Write

Kindling

 

 

The biggies…. The following strategies require a little more planning to use. They are all very effective. Your school has folders and materials that specifically explain these strategies.

 

Word Works-Cracking Vocabulary’s CODE – 4 phases of vocabulary learning

  • Connect – new words to prior knowledge
  • Organize – new words to find relationships
  • Deep-processing – internalizing the words, deepen the understanding
  • Exercise – or rehearse their knowledge of new words

 

Reading for Meaning – strategy that helps students become proficient at making claims, finding main ideas, and using reasoning and details to support their ideas.

·         Students are presented with a series of statements before they read the text, they need to either agree/disagree with the statement. 

·         After reviewing the statements, the students read the text and collect evidence either for or against the statements.

 

Task Rotation – creating activities that fit students’ learning styles:

·         Mastery

·         Interpersonal

·         Understanding

·         Self-expressive

 

Interactive Lecture – strategy that increases the students’ ability to remember, comprehend, and think actively about a lectures’ content. It engages the students by moving them through the four phases:

  • Connect – hook students’ attention
  • Organize – use graphic organizer/note taking procedure to help organize info
  • Deep-processing – pause every 5-7 min during lecture to allow students time to process information with questions in the different learning styles
  • Exercise – use review questions and have students use notes in a synthesis or application task.

 

Activities/Tools

These activities or tools are easy to slip in anywhere within your unit plan. They can be used for the CODE strategies, class openers, to brainstorm, to review, or as energizers for those “glazed over” moments. They are categorized somewhat, but several of these activities can be used in more than one category. Many of the strategies are referenced (in parenthesis) if you want more information on each of these activities.

 

Class Openers

Fact or Fiction/ Three’s a Crowd – Students decide which word/fact of three doesn’t belong and explain why. (Tool book p.10)

Anticipation Guides – Teacher prepares 3-5 statements based on the content that the students will be reading. Students are asked to decide which statements they believe the text will support. Teacher develops a class tally for each statement and discusses opinions. Students then read text. (Tool book p. 40)

Give one, Get one- students generate ideas from a question posed by the teacher, then have to collect a predetermined number of ideas from their classmates. (Tool book, p.11)

KWL- Tool to assess students’ prior knowledge, help generate questions about what they want to learn, and encourage reflection about what they have learned. (Tool book, p.28)

Spider List/Fist list/Fishbone- teacher provides a category in palm of hand/belly of spider and the students brainstorm ideas to fill in the fingers/legs/bones of hand print, spider, or fish sketch or vice versa.

Word Association/3-way tie- students select 3 words from a unit vocabulary and arrange them in a triangle. They then connect the words with lines then write a sentence that describes the relationship between the words that are connected. (binder)

 

Content Teaching

Word Wall – Collection of words on the wall for students to use during their reading and writing (Binder)

Reciprocal Learning/Peer Practice Strategy- students work in pairs (player and coach) to review or read & summarize concepts.

Think/pair/share-teacher poses a question, the students think and construct a response, then share their ideas with a neighbor, teacher records/collects their ideas. (Tool book, p.10)

Vocabulary Notebook- Where students collect critical vocabulary In the notebook students can write their initial “educated” definitions, then they can write the dictionary definition, and maybe a visual image as well. There is a lot of variations to this one. (Tool book, p. 92; binder)

Group & Labeling – students examine a list of vocabulary words and place them into groups based on common characteristics. For each group that students create, they devise a label that describes

Etch-a-sketch – students draw pictures, symbols, or icons to represent ideas presented in a lecture, reading, or other form of presentation (Tool book, p. 60)

Collaborative Summarizing – After lecture or reading, the students are asked to identify the 3-6 most important ideas. Students then pair up and compare their lists and come up with a consensus of the most important ideas with their partner. They (the group of two) pair up with another group of two and compare lists and once again come with an agreed upon list of 3-6 important ideas. These four use their list to create a collaborative summary. (Tool book, p.78)

Jigsaw- students work cooperatively with each student having an assigned task within the group to accomplish/perform.

Carousel Brainstorming – teacher generates different styles of questions & posts them around the room. The students work in groups of 3-5, rotates around the room to reading the question, the other responses, and either expands on existing ideas or develops a new idea. (Tool book, p.19)

4-2-1 Free Write – students identify 4 important ideas previously presented in the lesson. Each student meets with another student to compare ideas and decide on the two most important from their lists of four. This pair meets with another pair. They compare their ideas, then come to a consensus on the most important idea. The students then take this and do a free write, explaining all they know about the big idea. (Tool book, p. 82)

Concept Attainment – teacher presents examples and non-examples of a concept in a class discussion, the students use these to brainstorm the key attributes/characteristics of the concept.

Compare/Contrast – comparing likenesses and differences. The Georgia website I sent you has several different variations on this that are interesting.

Kindling –   F - Find a question that can be explored

              I  - Internalize the question

              R - Record your thoughts (sketch, write…)

              E - Exchange ideas with a neighbor

              S - Select and record your ideas in public (Tool Book p.74)

Comprehension Menu – an abbreviated version of Task Rotation. Teacher creates at least four questions in the four learning styles about the content. (Tool book, 162)

 

 

Review Activities/Tools

1,2,3,4 – teacher stops 5 minutes before end of class period to allow the students to reflect upon what was presented by writing in this format:

        1 – What was the big idea

        2 – Important details discussed

        3 – Personal connections discovered

        4 – Questions students have about the content

Boggle – students review notes for 2 minutes, then list as many ideas or details they can remember for 2-5 minutes, then students share their ideas with 1 or 2 other students and can add to their lists. Lastly, students pair up and compete with another student using the Boggle technique (They earn a point for every idea that their Boggle partner doesn’t have). Then the students go back to their study teams and compute the team score. (Tool book p.134)

Memory Box – a box usually put on a test where students can take the first five minutes to list as many things they can remember(formulas, definitions, etc…)

TGT – (Teams Game Tournament) teacher creates vocabulary/fact cards. The teacher divides the students heterogeneously by academic ability. This is the study group. After they have studied for a while, they then move to homogenous groups established by the teacher and compete against each other. They follow the points system to see how many points they take back to their study teams. This is a great review activity, it takes some time to prepare it, but it is well worth it.  

Jeopardy - This follows the same format as the game show. It is great to use on the active board. There are many already developed on the Ashland Schools website.

Categories – technique for forming groups and reviewing content. (Tool book,

p. 138)

 

Tool Book refers to:  Silver & Strong, 2001 Tools for Promoting Active, In-Depth Learning, Thoughtful Education Press.

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